A theory of diffuse X-ray scattering from a semiconductor system with ellipsoidal quantum dots (QDs) has been developed. The elastic strains outside a QD are calculated using the method of multipole expansions. An expression for the lattice displacement field is presented accurate to within the quadrupole term of expansion. Using the proposed approach, an analytical solution for the diffuse scattering from a crystalline medium with ellipsoidal inclusions is obtained. Reciprocal-space maps of the scattering intensity distribution are obtained by numerical calculations for QDs with various ratios of the height to lateral radius.